Alphabets, rewriting trails and periodic representations in algebraic bases

For β> 1 a real algebraic integer (the base), the finite alphabets A⊂ Z which realize the identity Q(β) = Per _A(β) , where Per _A(β) is the set of complex numbers which are (β, A) -eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base β and Lehmer’s problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in Q(β) , generalizing Schmidt’s theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases γs:=γn,m1,…,ms such that γs-1 is the unique root in (0, 1) of an almost Newman polynomial of the type -1+x+xn+xm1+⋯+xms, n≥ 3 , s≥ 1 , m₁- n≥ n- 1 , m_q+1- m_q≥ n- 1 for all q≥ 1. For β> 1 a reciprocal algebraic integer close to one, the poles of modulus < 1 of the dynamical zeta function of the β-shift ζ_β(z) are shown, under some assumptions, to be zeroes of the minimal polynomial of β.